Krein's Theorem Research Articles

On Location of the Matrix Spectrum with Respect to a Parabola

In the present article, we consider the problem on location of the matrix spectrum withrespect to a parabola. In terms of solvability of a matrix Lyapunov type equation, we provetheorems on location of the matrix spectrum in certain domains {cal P}_i (bounded by a parabola) and {cal P}_e (lying outside the closure of {cal P}_i). A solution to the matrix equation is constructed.We use this equation and prove an analog of the Lyapunov–Krein theorem on dichotomy ofthe matrix spectrum with respect to a parabola.

Friedel formula and Krein's theorem in complex potential scattering theory

In this work, the generalization of the Friedel formula and Krein's theorem in complex potential scattering theory is presented. The consequences of various symmetry constraints on dynamical systems are discussed. In addition, the Muskhelishvili-Omn\`es representation of Krein's theorem is also given and discussed.

Non‐classical Tauberian and Abelian type criteria for the moment problem

AbstractThe aim of this paper is to provide some new criteria for the determinacy problem of the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non‐classical Tauberian results for moment sequences that are analogues to the ones developed in [8] and [3] for the bi‐lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time‐dependent moment problem for the law of log‐Lévy processes viewed as a generalization of the log‐normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally‐negative Lévy processes having a Gaussian component, which may be of independent interest.

The floating-body problem: an integro-differential equation without irregular frequencies

The linear boundary value problem under consideration describes time-harmonic motion of water in a horizontal three-dimensional layer of constant depth in the presence of an obstacle adjacent to the upper side of the layer (floating body). This problem for a complex-valued harmonic function involves mixed boundary conditions and a radiation condition at infinity. Under rather general geometric assumptions the existence of a unique solution is proved for all values of the nonnegative problem's parameter related to the frequency of oscillations. The proof is based on the representation of solution as a sum of simple- and double-layer potentials with densities distributed over the obstacle's surface, thus reducing the problem to an indefinite integro-differential equation. The latter is shown to be soluble for all continuous right-hand side terms for which purpose S.~G. Krein's theorem about indefinite equations is used.

Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups

This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica–Speicher quantum groups and we list some open problems.

A Formula for the Fréchet Derivative of a Generalized Matrix Function

We state and prove an analogue of the Daleckii--Krein theorem, thus obtaining an explicit formula for the Fréchet derivative of generalized matrix functions. Moreover, we prove the differentiability of generalized matrix functions of real matrices under very mild assumptions. For complex matrices, we argue that, under the same assumptions, generalized matrix functions are real-differentiable but generally not complex-differentiable. Finally, we discuss the application of our results to the study of the condition number of generalized matrix functions. Along our way, we also derive generalized matrix functional analogues of a few classical theorems on polynomial interpolation of classical matrix functions and their derivatives.

Best Rational Approximation of Functions with Logarithmic Singularities

We consider functions \(\omega \) on the unit circle \({\mathbb T}\) with a finite number of logarithmic singularities. We study the approximation of \(\omega \) by rational functions and find an asymptotic formula for the distance in the \({{\mathrm{BMO}}}\)-norm between \(\omega \) and the set of rational functions of degree n as \(n\rightarrow \infty \). Our approach relies on the Adamyan–Arov–Krein theorem and on the study of the asymptotic behavior of singular values of Hankel operators.

Krein's Theorem for the Dissipative Operators with Finite Impulsive Effects

In this article, singular dissipative operators with finite impulsive conditions are investigated. In particular, after passing to the inverse operators, it is obtained that the imaginary parts of the inverse operators are nuclear. Finally, using Krein's theorem, it is proved that all root vectors of the singular dissipative operators with finite impulsive conditions are complete in the Hilbert space.

Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

In this paper we prove two consequences of the subnormal character of the Hessenberg matrix D when the hermitian matrix M of an inner product is a moment matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We also prove an extension of the Krein theorem for discrete measures on the complex plane based on properties of subnormal operators.

On Krein-Like Theorems for Noncanonical Hamiltonian Systems with Continuous Spectra: Application to Vlasov-Poisson

The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f 0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f 0. We prove that for each f 0 there is an arbitrarily small δf′0 in such that f 0+δf 0 is unstable. When f 0 is perturbed by an area preserving rearrangement, f 0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in Morrison and Pfirsch (1992) and Morrison (2000). If there is a signature change, then there is a rearrangement of f 0 that is unstable and arbitrarily close to f 0 with f′0 in W.1,1 This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in Cn norm that makes f 0 unstable. If f 0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.

Representation and index theory for Toeplitz operators

We study Toeplitz operators on the Hardy spaces of connected compact abelian groups and of tube-type bounded symmetric domains. A representation theorem for these operators and for classes of abstract Toeplitz elements in C*-algebras is proved. This is used to give a unified treatment to index theory in this setting, and a variety of new index theorems are proved that generalize the Gohberg–Krein theorem for Toeplitz operators on the Hardy space of the unit circle in the plane.

Unfolding the conical zones of the dissipation-induced subcritical flutter for the rotationally symmetrical gyroscopic systems

Flutter of an elastic body of revolution spinning about its axis of symmetry is prohibited in the subcritical spinning speed range by the Krein theorem for the Hamiltonian perturbations. Indefinite damping creates conical domains of the subcritical flutter (subcritical parametric resonance) bifurcating into the pockets of two Whitney's umbrellas when non-conservative positional forces are additionally taken into account. This explains why in contrast to the common intuition, but in agreement with experience, symmetry-breaking stiffness variation can promote subcritical friction-induced oscillations of the rotor rather than inhibit them.

A Generalization of the One-Step Theorem for Matrix Polynomials

In this paper we give a new proof of Krein's Theorem for orthogonal matrix polynomials based on a "one-step" version of the theorem. This parallels the proof given in [3] of Krein's Theorem in the scalar case.

Extremal positive and self-adjoint extensions of suboperators

F. ollowing Halmos [1] a suboperator is a map from a subspace of a (complex) Hilbert space into the whole space which is a restriction of some bounded linear transformation, an operator on the space. A suboperator is said to be subpositive, subself-adjoint (and so on) if it is a restriction of a positive, self-adjoint (and so on) operator of the space. The characterization problem of subself-adjoint suboperators is solved by Krein, see [2]. Krein shows that a bounded and symmetric linear map from a subspace of a Hilbert space has a symmetric extension to the whole space (hence a self-adjoint extension) with the same bound. Krein's method of proof yields the normpreserving smallest, resp. largest self-adjoint extensions, in the usual ordering of self-adjoint operators on Hilbert space, too. The extension theorem for subpositive suboperators, due to one of the authors [3], led naturally to the reduction of Krein's theorem to the just mentioned positive extendibility problem. Another characterization of subpositive suboperators is given by Halmos [1]. The purpose of this note is first to show that the extension of a subpositive suboperator constructed in [3] is of minimal norm and smallest in the ordering of self-adjoint operators, between all the positive extensions. The existence of a largest positive extension (with the same norm) is then a simple consequence of the existence of the smallest positive extension. The extremal norm-preserving extension problem with respect to subself-adjoint suboperators is thus reduced to the corresponding, just mentioned question for subpositive suboperators.

A quantitative maximum entropy theorem for the real line

A bound B is constructed for a large and natural class of entropy integrals on the real line. The construction involves Burg's maximum entropy theorem for the circle group and techniques from harmonic analysis. As the solution of an extension problem, the construction of B leads to a further construction problem which can be viewed as a means of refining Krein's theorem on positive definite extensions.

Vector variant of the Adamyan—Arov—Krein theorem

Vector variant of the Adamyan—Arov—Krein theorem

Representability problem in quantum mechanics

The notion of reduced density operators is generalized and placed in the context of abstract operator algebras. In this setting it is shown that the representability problem is one of finding positive extensions to positive linear functionals, a problem that has been addressed by M. G. Krein in the 1930s. An extension of Krein's theorem is given and used in the analysis of the fermion representability problem.

Scattering theory and cluster calculations

It is shown that a formula due to Lloyd (1967) for the density of states of a cluster of muffin-tin potentials can be derived very easily from a theorem in formal scattering theory that was proved by Krein (1953). Some questions in the usual interpretation of the formulae are discussed in the light of Krein's theorem and also the time-delay operator. Generalised Jost functions are introduced and used to furnish a picture of the limiting process in various kinds of cluster calculations.

Locally convex topologies and the convex compactness property

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

A convergence theorem with boundary

This paper contains a bounded-conver gence type theorem that depends on the fact that certain functions attain their suprema. Among the applications discussed are Rainwater's theorem and two technical results, one used in the proof of the Choquet-Bishop-deLeeuw theorem and the other in the proof of Krein's Theorem.